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Theorem ablfac1c 18293

Description: The factors of ablfac1b 18292 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
ablfac1.b	⊢ 𝐵 = (Base‘𝐺)
ablfac1.o	⊢ 𝑂 = (od‘𝐺)
ablfac1.s	⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
ablfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
ablfac1.f	⊢ (𝜑 → 𝐵 ∈ Fin)
ablfac1.1	⊢ (𝜑 → 𝐴 ⊆ ℙ)
ablfac1c.d	⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}
ablfac1.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)

**Assertion**
Ref	Expression
ablfac1c	⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Distinct variable groups: 𝑤,𝑝,𝑥,𝐵 𝐷,𝑝,𝑥 𝜑,𝑝,𝑤,𝑥 𝐴,𝑝,𝑥 𝑂,𝑝,𝑥 𝐺,𝑝,𝑥 Allowed substitution hints: 𝐴(𝑤) 𝐷(𝑤) 𝑆(𝑥,𝑤,𝑝) 𝐺(𝑤) 𝑂(𝑤)

Proof of Theorem ablfac1c Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablfac1.f	. 2 ⊢ (𝜑 → 𝐵 ∈ Fin)
2		ablfac1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	dprdssv 18238	. . 3 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝐵)
5		ssfi 8065	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵) → (𝐺 DProd 𝑆) ∈ Fin)
6	1, 3, 5	sylancl 693	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ Fin)
7		hashcl 13009	. . . . 5 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → (#‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∈ ℕ₀)
9		hashcl 13009	. . . . 5 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ₀)
10	1, 9	syl 17	. . . 4 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
11		ablfac1.o	. . . . . . 7 ⊢ 𝑂 = (od‘𝐺)
12		ablfac1.s	. . . . . . 7 ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})
13		ablfac1.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Abel)
14		ablfac1.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℙ)
15	2, 11, 12, 13, 1, 14	ablfac1b 18292	. . . . . 6 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
16		dprdsubg 18246	. . . . . 6 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
18	2	lagsubg 17479	. . . . 5 ⊢ (((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (#‘(𝐺 DProd 𝑆)) ∥ (#‘𝐵))
19	17, 1, 18	syl2anc 691	. . . 4 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∥ (#‘𝐵))
20		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑞 → (𝑤 ∥ (#‘𝐵) ↔ 𝑞 ∥ (#‘𝐵)))
21		ablfac1c.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}
22	20, 21	elrab2 3333	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐷 ↔ (𝑞 ∈ ℙ ∧ 𝑞 ∥ (#‘𝐵)))
23		ablfac1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
24	23	sseld 3567	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ 𝐷 → 𝑞 ∈ 𝐴))
25	22, 24	syl5bir 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝑞 ∈ ℙ ∧ 𝑞 ∥ (#‘𝐵)) → 𝑞 ∈ 𝐴))
26	25	impl 648	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (#‘𝐵)) → 𝑞 ∈ 𝐴)
27	2, 11, 12, 13, 1, 14	ablfac1a 18291	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑆‘𝑞)) = (𝑞↑(𝑞 pCnt (#‘𝐵))))
28		fvex 6113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘𝐺) ∈ V
29	2, 28	eqeltri 2684	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 ∈ V
30	29	rabex 4740	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))} ∈ V
31	30, 12	dmmpti 5936	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = 𝐴
32	31	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑆 = 𝐴)
33	15, 32	dprdf2 18229	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺))
34	33	ffvelrnda 6267	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘𝐺))
35	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐺dom DProd 𝑆)
36	31	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → dom 𝑆 = 𝐴)
37		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
38	35, 36, 37	dprdub 18247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))
39	17	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
40		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ↾_s (𝐺 DProd 𝑆)) = (𝐺 ↾_s (𝐺 DProd 𝑆))
41	40	subsubg 17440	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → ((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ↔ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))))
42	39, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ↔ ((𝑆‘𝑞) ∈ (SubGrp‘𝐺) ∧ (𝑆‘𝑞) ⊆ (𝐺 DProd 𝑆))))
43	34, 38, 42	mpbir2and 959	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
44	40	subgbas 17421	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
45	39, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))))
46	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝐺 DProd 𝑆) ∈ Fin)
47	45, 46	eqeltrrd 2689	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin)
48		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) = (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))
49	48	lagsubg 17479	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑞) ∈ (SubGrp‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∧ (Base‘(𝐺 ↾_s (𝐺 DProd 𝑆))) ∈ Fin) → (#‘(𝑆‘𝑞)) ∥ (#‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
50	43, 47, 49	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑆‘𝑞)) ∥ (#‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
51	45	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝐺 DProd 𝑆)) = (#‘(Base‘(𝐺 ↾_s (𝐺 DProd 𝑆)))))
52	50, 51	breqtrrd 4611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑆‘𝑞)) ∥ (#‘(𝐺 DProd 𝑆)))
53	27, 52	eqbrtrrd 4607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞↑(𝑞 pCnt (#‘𝐵))) ∥ (#‘(𝐺 DProd 𝑆)))
54	14	sselda 3568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ ℙ)
55	8	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∈ ℤ)
56	55	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝐺 DProd 𝑆)) ∈ ℤ)
57		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
58		ablgrp 18021	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
59	2	grpbn0 17274	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
60	13, 58, 59	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ ∅)
61		hashnncl 13018	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
62	1, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
63	60, 62	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ)
64	63	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (#‘𝐵) ∈ ℕ)
65	57, 64	pccld 15393	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (#‘𝐵)) ∈ ℕ₀)
66	54, 65	syldan 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (#‘𝐵)) ∈ ℕ₀)
67		pcdvdsb 15411	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℙ ∧ (#‘(𝐺 DProd 𝑆)) ∈ ℤ ∧ (𝑞 pCnt (#‘𝐵)) ∈ ℕ₀) → ((𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))) ↔ (𝑞↑(𝑞 pCnt (#‘𝐵))) ∥ (#‘(𝐺 DProd 𝑆))))
68	54, 56, 66, 67	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))) ↔ (𝑞↑(𝑞 pCnt (#‘𝐵))) ∥ (#‘(𝐺 DProd 𝑆))))
69	53, 68	mpbird 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
70	69	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∈ 𝐴) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
71	26, 70	syldan 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (#‘𝐵)) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
72		pceq0 15413	. . . . . . . . . 10 ⊢ ((𝑞 ∈ ℙ ∧ (#‘𝐵) ∈ ℕ) → ((𝑞 pCnt (#‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (#‘𝐵)))
73	57, 64, 72	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → ((𝑞 pCnt (#‘𝐵)) = 0 ↔ ¬ 𝑞 ∥ (#‘𝐵)))
74	73	biimpar 501	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (#‘𝐵)) → (𝑞 pCnt (#‘𝐵)) = 0)
75		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
76	75	subg0cl 17425	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺) → (0_g‘𝐺) ∈ (𝐺 DProd 𝑆))
77		ne0i 3880	. . . . . . . . . . . . . 14 ⊢ ((0_g‘𝐺) ∈ (𝐺 DProd 𝑆) → (𝐺 DProd 𝑆) ≠ ∅)
78	17, 76, 77	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≠ ∅)
79		hashnncl 13018	. . . . . . . . . . . . . 14 ⊢ ((𝐺 DProd 𝑆) ∈ Fin → ((#‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
80	6, 79	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((#‘(𝐺 DProd 𝑆)) ∈ ℕ ↔ (𝐺 DProd 𝑆) ≠ ∅))
81	78, 80	mpbird 246	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) ∈ ℕ)
82	81	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (#‘(𝐺 DProd 𝑆)) ∈ ℕ)
83	57, 82	pccld 15393	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (#‘(𝐺 DProd 𝑆))) ∈ ℕ₀)
84	83	nn0ge0d 11231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → 0 ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
85	84	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (#‘𝐵)) → 0 ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
86	74, 85	eqbrtrd 4605	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ ¬ 𝑞 ∥ (#‘𝐵)) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
87	71, 86	pm2.61dan 828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ℙ) → (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
88	87	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℙ (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆))))
89	10	nn0zd 11356	. . . . . 6 ⊢ (𝜑 → (#‘𝐵) ∈ ℤ)
90		pc2dvds 15421	. . . . . 6 ⊢ (((#‘𝐵) ∈ ℤ ∧ (#‘(𝐺 DProd 𝑆)) ∈ ℤ) → ((#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆)))))
91	89, 55, 90	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)) ↔ ∀𝑞 ∈ ℙ (𝑞 pCnt (#‘𝐵)) ≤ (𝑞 pCnt (#‘(𝐺 DProd 𝑆)))))
92	88, 91	mpbird 246	. . . 4 ⊢ (𝜑 → (#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)))
93		dvdseq 14874	. . . 4 ⊢ ((((#‘(𝐺 DProd 𝑆)) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) ∧ ((#‘(𝐺 DProd 𝑆)) ∥ (#‘𝐵) ∧ (#‘𝐵) ∥ (#‘(𝐺 DProd 𝑆)))) → (#‘(𝐺 DProd 𝑆)) = (#‘𝐵))
94	8, 10, 19, 92, 93	syl22anc 1319	. . 3 ⊢ (𝜑 → (#‘(𝐺 DProd 𝑆)) = (#‘𝐵))
95		hashen 12997	. . . 4 ⊢ (((𝐺 DProd 𝑆) ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘(𝐺 DProd 𝑆)) = (#‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
96	6, 1, 95	syl2anc 691	. . 3 ⊢ (𝜑 → ((#‘(𝐺 DProd 𝑆)) = (#‘𝐵) ↔ (𝐺 DProd 𝑆) ≈ 𝐵))
97	94, 96	mpbid 221	. 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) ≈ 𝐵)
98		fisseneq 8056	. 2 ⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd 𝑆) ⊆ 𝐵 ∧ (𝐺 DProd 𝑆) ≈ 𝐵) → (𝐺 DProd 𝑆) = 𝐵)
99	1, 4, 97, 98	syl3anc 1318	1 ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ≈ cen 7838 Fincfn 7841 0cc0 9815 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ↑cexp 12722 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 pCnt cpc 15379 Basecbs 15695 ↾_s cress 15696 0_gc0g 15923 Grpcgrp 17245 SubGrpcsubg 17411 odcod 17767 Abelcabl 18017 DProd cdprd 18215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-eqg 17416 df-ghm 17481 df-gim 17524 df-ga 17546 df-cntz 17573 df-oppg 17599 df-od 17771 df-lsm 17874 df-pj1 17875 df-cmn 18018 df-abl 18019 df-dprd 18217

This theorem is referenced by: ablfaclem2 18308

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